$ \acute{C} $iri$ \acute{c} $-Reich-Rus type weakly contractive mappings and related fixed point results in modular-like spaces with application

Fatemeh Lael; Naeem Saleem; Işık Hüseyin; Manuel de la Sen; Fatemeh Lael; Naeem Saleem; Işık Hüseyin; Manuel de la Sen

doi:10.3934/math.2022898

AIMS Mathematics

2022, Volume 7, Issue 9: 16422-16439. doi: 10.3934/math.2022898

Previous Article Next Article

Research article

$ \acute{C} $iri$ \acute{c} $-Reich-Rus type weakly contractive mappings and related fixed point results in modular-like spaces with application

1.
Department of Mathematics, Buein Zahra Technical University, Buein Zahra, Qazvin, Iran
2.
Department of Mathematics, University of Management and Technology, Lahore, Pakistan
3.
Department of Engineering Science, Bandırma Onyedi Eylül University, Bandırma 10200, Balıkesir, Turkey
4.
Institute of Research and Development of Processes, Campus of Leioa (Bizkaia), University of the Basque Country, Barrio Sarriena, P.O. Box 644-Bilbao, 48940 Leioa, Spain

Received: 05 April 2022 Revised: 27 June 2022 Accepted: 27 June 2022 Published: 06 July 2022
MSC : 46E30, 47H10, 54H25

In this article, we define a new space named the modular-like space with its related concepts to prove the existence of a fixed point and a point of coincidence for mappings on this space. Also, we defined $ \acute{C} $iri$ \acute{c} $-Reich-Rus type weakly contractive mappings on modular-like spaces and discussed some conditions that guarantee the existence of the fixed points for these kind of mappings. Some examples are also provided to elaborate the usability of our main results. It is worth mentioning that a modular-like space is a generalization of a modular space, thus our theorems are more general and applicable than the fixed point theorems on modular spaces.
- modular space,
- fixed point,
- point of coincidence,
- integral equation
Citation: Fatemeh Lael, Naeem Saleem, Işık Hüseyin, Manuel de la Sen. $ \acute{C} $iri$ \acute{c} $-Reich-Rus type weakly contractive mappings and related fixed point results in modular-like spaces with application[J]. AIMS Mathematics, 2022, 7(9): 16422-16439. doi: 10.3934/math.2022898

Related Papers:

Abstract

In this article, we define a new space named the modular-like space with its related concepts to prove the existence of a fixed point and a point of coincidence for mappings on this space. Also, we defined $ \acute{C} $iri$ \acute{c} $-Reich-Rus type weakly contractive mappings on modular-like spaces and discussed some conditions that guarantee the existence of the fixed points for these kind of mappings. Some examples are also provided to elaborate the usability of our main results. It is worth mentioning that a modular-like space is a generalization of a modular space, thus our theorems are more general and applicable than the fixed point theorems on modular spaces.

References

[1]	M. Abbas, D. Dori$\acute{c}$, Common fixed point theorem for four mappings satisfying generalized weak contractive condition, Filomat, 24 (2010), 1–10. https://doi.org/10.2298/FIL1002001A doi: 10.2298/FIL1002001A
[2]	A. A. Abdo, M. A. Khamsi, On common fixed points in modular vector spaces, Fixed Point Theory Appl., 229 (2015). https://doi.org/10.1186/s13663-015-0478-z
[3]	A. Ait Taleb, Hanebaly, A fixed point theorem and its application to integral equations in modular function spaces, Proc. Amer. Math. Soc., 128 (1999), 419–426.
[4]	S. A. Al-Mezel, A. Al-Roqi, M. A. Khamsi, One-local retract and common fixed point in modular function spaces, Fixed Point Theory Appl., 109 (2012). https://doi.org/10.1186/1687-1812-2012-109
[5]	A. H. Ansari, T. Došenovic, S. Radenovi$\acute{c}$, N. Saleem, V. Šešum-Cavic, J. Vujakovic, $C$-class functions on some fixed point results in ordered partial metric spaces via admissible mappings, Novi Sad J. Math., 49 (2019), 101–116. https://doi.org/10.30755/NSJOM.07794 doi: 10.30755/NSJOM.07794
[6]	A. H. Ansari, J. M. Kumar, N. Saleem, Inverse-$C$-class function on weak semi compatibility and fixed point theorems for expansive mappings in $G$-metric spaces, Math. Morav., 24 (2020), 93–108. https://doi.org/10.5937/MatMor2001093H doi: 10.5937/MatMor2001093H
[7]	H. Aydi, C. M. Chen, E. Karapinar, Interpolative $\acute{C}$iri$\acute{c}$-Reich-Rus type contractions via the Branciari distance, Mathematics, 7 (2019), 84. https://doi.org/10.3390/math7010084 doi: 10.3390/math7010084
[8]	I. A. Bakhtin, The contraction mapping principle in almost metric spaces, Funct. Anal., Unianowsk, Gos. Ped. Inst., 30 (1989), 26–37.
[9]	S. Banach, Sur les op$\acute{e}$rations dans les ensembles abstraits et leur application aux$\acute{e}$quations intégrales, Fund. Math., 3 (1922), 133–181.
[10]	P. Debnath, N. Konwar, S. Radenovi$\acute{c}$, Metric fixed point theory: Applications in science, engineering and behavioural sciences, Singapore, Springer, 2021. https://doi.org/10.1007/978-981-16-4896-0
[11]	S. Dhompongsa, T. Dominguez-Benavides, A. Kaewcharoen, B. Panyanak, Fixed point theorems for multivalued mappings in modular function spaces, Sci. Math. Jpn., 63 (2006), 161–169.
[12]	T. Dominguez-Benavides, M. A. Khamsi, S. Samadi, Asymptotically regular mappings in modular function spaces, Sci. Math. Jpn., 2 (2001), 295–304.
[13]	D. Doric, Common fixed point for generalized $(\psi-\phi)$-weak contractions, Appl. Math. Lett., 22 (2009), 1896–1900. https://doi.org/10.1016/j.aml.2009.08.001 doi: 10.1016/j.aml.2009.08.001
[14]	Y. Errai, E. M. Marhrani, M. Aamri, Fixed points of $g$-interpolative $\acute{C}$iri$\acute{c}$-Reich-Rus type contractions in $b$- metric spaces, Axioms, 9 (2020), 132. https://doi.org/10.3390/axioms9040132 doi: 10.3390/axioms9040132
[15]	Y. Errai, E. M. Marhrani, M. Aamri, Related fixed point theorems in partially ordered $b$-metric spaces and applications to integral equations, Abst. Appl. Anal., 2021 (2021), 6672724. https://doi.org/10.1155/2021/6672724 doi: 10.1155/2021/6672724
[16]	R. Kannan, Some results on fixed points, Bull. Cal. Math. Soc., 60 (1968), 71-76.
[17]	E. Karapinar, Revisiting the Kannan type contractions via interpolation, Adv. Theory Nonlinear Anal. Appl., 2 (2018), 85-87.
[18]	E. Karapinar, O. Alqahtani, H. Aydi, On interpolative Hardy-Rogers type contractions, Symmetry, 11 (2019), 8. https://doi.org/10.3390/sym11010008 doi: 10.3390/sym11010008
[19]	S. Karthikeyan, C. Park, P. Palani, T. R. K. Kumar, Stability of an additive-quartic functional equation in modular spaces, J. Math. Comput. Sci., 26 (2022), 22–40. http://dx.doi.org/10.22436/jmcs.026.01.04 doi: 10.22436/jmcs.026.01.04
[20]	M. A. Khamsi, A convexity property in modular function spaces, Math. Jpn., 44 (1996), 269–279.
[21]	M. A. Khamsi, W. M. Kozlowski, On asymptotic pointwise nonexpansive mappings in modular function spaces, J. Math. Anal. Appl., 380 (2011), 697–708. https://doi.org/10.1016/j.jmaa.2011.03.031 doi: 10.1016/j.jmaa.2011.03.031
[22]	M. A. Khamsi, W. M. Kozlowski, S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal., 14 (1990), 935–953. https://doi.org/10.1016/0362-546X(90)90111-S doi: 10.1016/0362-546X(90)90111-S
[23]	M. A. Khamsi, W. M. Kozlowski, C. Shutao, Some geometrical properties and fixed point theorems in Orlicz spaces, J. Math. Anal. Appl., 155 (1991), 393–412. https://doi.org/10.1016/0022-247X(91)90009-O doi: 10.1016/0022-247X(91)90009-O
[24]	W. M. Kozlowski, Modular function spaces, New York, Marce Dekker Inc., 1988.
[25]	K. Kuaket, P. Kumam, Fixed points of asymptotic pointwise contractions in modular spaces, Appl. Math. Lett., 24 (2011), 1795–1798. https://doi.org/10.1016/j.aml.2011.04.035 doi: 10.1016/j.aml.2011.04.035
[26]	P. Kumam, Fixed point theorem for non-expansive mappings in modular spaces, Arch. Math., 40 (2004), 345–353.
[27]	F. Lael, K. Nourouzi, Fixed points of mappings defined on probabilistic modular spaces, Bull. Math. Anal. Appl., 4 (2012), 23–28.
[28]	F. Lael, K. Nourouzi, On the fixed points of correspondences in modular spaces, Int. Sch. Res. Notices, 2011 (2011), 530254. https://doi.org/10.5402/2011/530254 doi: 10.5402/2011/530254
[29]	F. Lael, S. Shabanian, Convexity and boundedness relaxation for fixed point theorems in modular spaces, Appl. Gen. Topol., 22 (2021), 91–108. https://orcid.org/0000-0003-3665-3401
[30]	Z. D. Mitrovi$\acute{c}$, S. Radenovi$\acute{c}$, H. Aydi, A. A. Altasan, C. $\ddot{O}$zel, On two new approaches in modular spaces, Ital. J. Pure Appl. Math., 41 (2019), 679–690.
[31]	C. Mongkolkeha, P. Kumam, Common fixed points for generalized weak contraction mappings in modular spaces, Sci. Math. Jpn., 75 (2012), 117–127.
[32]	C. Mongkolkeha, P. Kumam, Fixed point and common fixed point theorems for generalized weak contraction mappings of integral type in modular spaces, Int. J. Math. Math. Sci., 2011 (2011), 705943. https://doi.org/10.1155/2011/705943 doi: 10.1155/2011/705943
[33]	C. Mongkolkeha, P. Kumam, Some fixed point results for generalized weak contraction mappings in modular spaces, Int. J. Anal., 2013 (2013), 247378. https://doi.org/10.1155/2013/247378 doi: 10.1155/2013/247378
[34]	J. Musielak, Orlicz spaces and modular spaces, Lect. Notes Math., Springer-Verlag, 1983.
[35]	J. Musielak, W. Orlicz, On modular spaces, Stud. Math., 18 (1959), 49–65.
[36]	H. Nakano, Modulared semi-ordered linear spaces, Maruzen, Tokyo, 1950.
[37]	G. A. Okeke, D. Francis, Fixed point theorems for asymptotically T-regular mappings in preordered modular G-metric spaces applied to solving nonlinear integral equations, J. Anal., 30 (2022), 501–545. https://doi.org/10.1007/s41478-021-00354-1 doi: 10.1007/s41478-021-00354-1
[38]	G. A. Okeke, S. H. Khan, Approximation of fixed point of multivalued $\rho$-quasi-contractive mappings in modular function spaces, Arab J. Math. Sci., 26 (2020), 75–93. https://doi.org/10.1016/j.ajmsc.2019.02.001 doi: 10.1016/j.ajmsc.2019.02.001
[39]	W. Orlicz, Über eine gewisse klasse von Räumen vom Typus B, Bull. Acad. Polon. Sci. A., 8-9 (1932), 207–220.
[40]	W. Orlicz, Über Räume $(L^M)$, Bull. Acad. Polon. Sci. A., 1936 (1936), 93–107.
[41]	O. Popescu, Fixed points for $(\psi, \phi)$-weak contractions, Appl. Math. Lett., 24 (2011), 1–4. https://doi.org/10.1016/j.aml.2010.06.024 doi: 10.1016/j.aml.2010.06.024
[42]	B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal., 47 (2001), 2683–2693. https://doi.org/10.1016/S0362-546X(01)00388-1 doi: 10.1016/S0362-546X(01)00388-1
[43]	I. A. Rus, Generalized contractions and applications, Cluj-Napoca, Cluj University Press, 2001.
[44]	N. Saleem, M. Abbas, B. Bin-Mohsin, S. Radenovi$\acute{c}$, Pata type best proximity point results in metric spaces, Miskolc Math. Notes, 21 (2020), 367–386.
[45]	N. Saleem, M. de la Sen, S. Farooq, Coincidence best proximity point results in Branciari metric spaces with applications, J. Funct. Space., 2020 (2020), 4126025. https://doi.org/10.1155/2020/4126025 doi: 10.1155/2020/4126025
[46]	N. Saleem, I. Iqbal, B. Iqbal, S. Radenović, Coincidence and fixed points of multivalued $F$-contractions in generalized metric space with application, J. Fixed Point Theory Appl., 22 (2020), 81. https://doi.org/10.1007/s11784-020-00815-3 doi: 10.1007/s11784-020-00815-3
[47]	W. Sintunavarat, Fixed point results in $b$-metric spaces approach to the existence of a solution for nonlinear integral equations, RACSAM, 110 (2016), 585–600. https://doi.org/10.1007/s13398-015-0251-5 doi: 10.1007/s13398-015-0251-5
[48]	D. Turkoglu, N. Manav, Fixed point theorems in a new type of modular metric spaces, Fixed Point Theory Appl., 2018 (2018), 25. https://doi.org/10.1186/s13663-018-0650-3 doi: 10.1186/s13663-018-0650-3
[49]	V. Todor$\check{c}$evi$\acute{c}$, Harmonic quasiconformal mappings and hyperbolic type metrics, Switzerland AG, Springer, 2019. https://doi.org/10.1007/978-3-030-22591-9

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)